When I discovered hypermetric inequalities (an attempt to characterize -embeddable metrics), there turned out to be applications in the geometry of numbers. I studied the generalization to quasi-metrics, hoping for similar applications.

1. Definition and examples

A quasi-metric is a nonnegative function on such that

;

, , , .

Example 1 Minkowski gauge of a compact convex set containing the origin.

A quasi-semi-metric is weightable iff its satisfies a cyclic inequality…

Any finite quasi-semi-metric is the shortest path quasi-semi-metric of a digraph. It is weighted iff the graph is na bidirectional tree.

Example 5 The hitting time quasi-metric is the expected number if step of random walk needed to go from to .

It is weightable. It is proportional to the effective resistance metric.

Example 6 Oriented multicut metrics.

An oriented multicut of a set is a partition with an ordering of the pieces. There is a corresponding quasi-semi-metric. It is weightable iff there are only two pieces.

1.2. Semantics of computation

Say a poset , having a smallest element, is dspo if each directed subset has a supremum. A Scott domain is a dspo where all sets are directed with supremum equal to , and each consistent has a supremum in .

All weighted quasi-semi-metrics on points form a polyhedral cone of dimension .

1.4. Derivation

-derivation is Gromov product with vertex , i.e.

[Makarychev, Makarychev]: Most weightable quasi-semi-metrics arise as Gromov product of some semi-metric.

Example 8 The -derivation of the -distance is called -quasi-metric and denoted by .

1.5. Generalization of -metrics

Theorem 1 A quasi-semi-metric embeds into iff it belongs to the cone generated by oriented cuts.

Note that an oriented cut quasi-semi-metric is the -derivation of the corresponding cut metric. It is weightable.

Example 9 embeds in if .

is the square of a -quasi-semi-metric.

Theorem 2 A quasi-semi-metric embeds into iff it is the quasi-semi-metric of subsets of some measure space.

What does not work: Hypercube does not have a natural quasi-semi-metric. Indeed, there are a lot of possible orientations on the hypercube. This is a whole subject in itself.

1.6. Hypermetric inequalities

Hypermetric inequalities describe certain facets of the cut cone.

Given integers which sum up to , metric satisfies

Say a metric is a hypermetric if it satisfies all these inequalities.

Given a lattice, and a maximum ball which contains no nonzero vectors of the lattice. It may contain lattice points on its boundary. Such a set is called a constellation. The square of Euclidean metric restricted to a constellation satisfies all hypermetric inequality. Conversely, every hypermetric is isometric to some constellation [Assouad-Deza].

This has a partial generalization to quasi-metrics. As yet, it looks rather ugly.